The Post-Brexit Identity Crisis

If you are going through withdrawal from loss of your EU citizenship, and horrified by your new status in your favorite EU country as a third-country national, to quote Douglas Adams, “Don’t Panic!”
 
Brexit presents you with the opportunity to obtain dual citizenship by applying for citizenship in an EU member country and becoming an EU citizen again.
 
Citizenship for a Specific EU Country
Each EU country has its own laws regarding the granting of citizenship.  Selecting the EU country whose citizenship you wish to acquire depends on why you want it and if you can qualify to obtain it.
 
Considerations
Landownership:  you may face restrictions as a non-citizen, depending on the specific country.
Residence:  there is no guarantee you will qualify for residence as a third-country national.
Employment:  there are fewer legal restrictions on employment for a citizen than a third-country national.
Local laws:  legal rights and privileges for a specific country that have particular value to you.
Family:  enabling family members to join you in country with fewer restrictions.
Travel:  freedom of movement within the Schengen Area to visit a country for up to 90 days in a 180 day period without needing to register your residence.
 
Three key questions to clarify your goal(s):
Do you want to visit EU countries?
Do you want to reside in a specific EU country?
Do you want to be an EU citizen?
 
You may decide after reviewing your goals, that the ability to simply travel to the EU on the UK passport is sufficient.  If, however it is not, you should research the citizenship requirements for your desired country.  There may be multiple methods of acquiring citizenship, specific to a given country.  You may be eligible for citizenship if your parents hold that citizenship or you are married to a citizen.  Or, you could apply for citizenship by naturalization.
 
Citizenship by Any EU Country
If you are more interested in obtaining EU citizenship than that of any specific EU country, and you have the financial means, you could simply obtain citizenship by investment from Bulgaria, Cyprus, or Malta.  All three countries allow dual citizenship as does the UK.  Professional consulting firms that assist prospective citizens include:
Arton Capital:  an authorized agent for the citizenship by investment programs for Bulgaria and Cyprus
La Vida Golden Visas: assistance with citizenship by investment programs for Cyprus and Malta.
Discus Holdings:  assistance with citizenship by investment programs for Cyprus and Malta.